Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 10 x^{2} - 5 x y + y^{2} + y z - z^{2} $ |
| $=$ | $20 x^{2} + 30 x y + 3 y^{2} - 2 y z + 2 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{3} y - 14 x^{2} y^{2} + 4 x^{2} z^{2} + 36 x y^{3} - 8 x y z^{2} + 181 y^{4} + 44 y^{2} z^{2} + 4 z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{5^2}\cdot\frac{532552812768647550000000xz^{11}-3033570527744299615800000xz^{9}w^{2}-14122553262591886738920000xz^{7}w^{4}+833629400388998627888000xz^{5}w^{6}+3333645105352424295624000xz^{3}w^{8}-100663521861984389600000xzw^{10}-71255721793003571812500y^{2}z^{10}+4372171448831824431729375y^{2}z^{8}w^{2}+8576506731889638149764500y^{2}z^{6}w^{4}-11455700017647457143929250y^{2}z^{4}w^{6}+953868102249594039146100y^{2}z^{2}w^{8}+1596562881608830901547y^{2}w^{10}-227768472732350542500000yz^{11}-2030020338704912989042500yz^{9}w^{2}+694971716600440923642000yz^{7}w^{4}+8334775552843389592187000yz^{5}w^{6}-2025839522691564172789600yz^{3}w^{8}+22231319902975265381132yzw^{10}+173119103417027075625000z^{12}-215827643642819180332500z^{10}w^{2}-3076283787697704371483250z^{8}w^{4}-1252722249834107720838000z^{6}w^{6}+288148315242575012123700z^{4}w^{8}+81584692869067270128268z^{2}w^{10}-299552123126744233394w^{12}}{3155868520110504000000xz^{11}+6404280598808042528000xz^{9}w^{2}+4659857498723421939200xz^{7}w^{4}+1364506329190686424960xz^{5}w^{6}+83555993434684823360xz^{3}w^{8}-18513044411950860160xzw^{10}-422256129143724870000y^{2}z^{10}-1772364078016766979200y^{2}z^{8}w^{2}-1640347384159675659520y^{2}z^{6}w^{4}-527256575286753239310y^{2}z^{4}w^{6}-27111353183561836820y^{2}z^{2}w^{8}+9515717191492051469y^{2}w^{10}-1349739097673188400000yz^{11}-1778305336806276143200yz^{9}w^{2}-691637402785598077920yz^{7}w^{4}+66149723602789401160yz^{5}w^{6}+100803960791290857184yz^{3}w^{8}+17269480257169200052yzw^{10}+1025890983212012300000z^{12}+1820554916275862043200z^{10}w^{2}+1062912717145605780520z^{8}w^{4}+159018617284201966520z^{6}w^{6}-44118681565274129556z^{4}w^{8}-8395708010431100876z^{2}w^{10}+1139893245960863146w^{12}}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.